Operations on Fuzzy Sets

Given ‘X’ to be universe of discourse, A and B are two fuzzy sets with membership function μA(x) and μB(x) then,


The union of two fuzzy sets A and B is a new fuzzy set A ∪ B also on ‘X’ with membership function defined as follow:

Membership Fuction for Union of Fuzzy Sets


Intersection of fuzzy sets A & B, is a new fuzzy set A ∩ B also on ‘X’ whose membership function is defined by

Intersection of Fuzzy Sets Membership Function


Compliment of a fuzzy set A is A with membership function

Compliment of Fuzzy Set A Membership Function

Product of Two Fuzzy Sets

The product of two fuzzy sets A & B is a new fuzzy set A.B with membership function:

membership function for the product of fuzzy sets


Two fuzzy sets A and B are said to be equal i.e, A = B if and only if μA(x) = μB(x) Which means their membership values must be equal.

Product of Fuzzy Sets with a Crisp Number

Multiplying a fuzzy set A by a crisp number ‘n’ results in a new fuzzy set n.A, whose membership function is

Product of fuzzy set with a crisp number

Power of a Fuzzy Set

The alpha power of a fuzzy set A is a new fuzzy set Aα whose membership function is:

Power of a Fuzzy Set Membership Function

that is, individual memberships power of α

Difference of Fuzzy Sets

The differences of two fuzzy sets A and B is a new fuzzy set A-B which is defined as

Difference of Fuzzy Sets A & B

Disjunctive Sum of A & B

It is the new fuzzy set defined as follow:

Disjoint Sum of A and B

Comments 1

  • A, assuming n is a member of the universal set of discourse, in no way shape or for yields a new set A' with u'(x) = n * u(x). It yields a set A'' with u''(x) = u(x * n). All the multiplication of a crisp function does is rescale (and potentially reverse, given that n is negative) the membership function. You're confusing the set of X with the set of F (fuzzy membership values). Additionally, the product of two fuzzy sets you're describing is actually an intersection operation in which the t-norm utilized is multiplication as opposed to the standard t-norm of min. The actual formula for the product of two fuzzy sets is far more complicated, and (for at least in the case of convex sets, I haven't worked much with non-convex) is u_{A * B}(w) = (for all z in X, sup_{z = x * y}(A(x),B(y)))(w). Your alpha power set is wrong, alpha is traditionally used as a "cutting function" on fuzzy sets, to eliminate support of a fuzzy set that is less than the alpha threshold, i.e. if A has a membership function of u(x), then A' = A^alpha has a membership function u'(x) = {u(x) where u(x) >= alpha, 0 otherwise}. The actual definition of the power of a fuzzy set is analogous to the multiplication operation I described above, but in the cases of even powers the support is usually restricted to positive values of the universe of discourse in order to remain analogous to crisp multiplication. What you're actually describing is a specific class of "linguistic hedges," which are functions on fuzzy sets designed to affect membership in a meaningful way. Your difference of fuzzy sets is correct, with the caveat that you should be calling it the "set difference" and be using the set difference operator as the arithmetic difference has a definition in line with the actual multiplication definition I described earlier, and in fact this arithmetic difference isn't actually defined in situations where the universe of discourse is limited to positive values. This is due to the potential of extending support into an undefined (i.e. negative) region.

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